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CS461 Artificial Intelligence © Pinar Duygulu Spring
2008
First Order Logic
Artificial Intelligence
Slides are mostly adapted from AIMA and MIT Open Courseware,
and Milos Hauskrecht (U. Pittsburgh)
2
Pros and cons of propositional logic
Propositional logic is declarative
Propositional logic allows partial/disjunctive/negated information– (unlike most data structures and databases)
Propositional logic is compositional:– meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2
Meaning in propositional logic is context-independent– (unlike natural language, where meaning depends on context)
Propositional logic has very limited expressive power– (unlike natural language)
– E.g., cannot say "pits cause breezes in adjacent squares“
• except by writing one sentence for each square
3
First Order Logic
4
First-order logic
• First-order logic (like natural language) assumes the world contains
– Objects: people, houses, numbers, colors, baseball games, wars, …
– Relations: red, round, prime, brother of, bigger than, part of, comes between, …
– Functions: father of, best friend, one more than, plus,
• (relations in which there is only one value for a given input)
5
FOL Motivation
6
Syntax of FOL: Basic elements
• Constants : KingJohn, 2, ...
• Predicates: Brother, >,...
• Functions : Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives , , , ,
• Equality =
• Quantifiers ,
7
Atomic sentences
Term = function (term1,...,termn)
or constant
or variable
Atomic sentence = predicate (term1,...,termn)
or term1 = term2
• E.g., Brother(KingJohn,RichardTheLionheart)
• > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))
8
Complex sentences
Complex sentences are made from atomic sentences using
connectives
S, S1 S2, S1 S2, S1 S2, S1 S2,
E.g.
Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)
>(1,2) ≤ (1,2)
>(1,2) >(1,2)
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Quantifiers
Universal quantification, (pronounced as “For all”)
x Cat(x) Mammal(x)
All cats are mammals
Existential quantification, (pronounced as “There exists”)
x Sister (x, Spot) Cat(x)
Spot has a sister who is a cat
• x P is true in a model m
iff P is true with x being each possible object in the model
• x P is true in a model m
iff P is true with x being some possible object in the model
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Truth in first-order logic
• Sentences are true with respect to a model and an interpretation
• Model contains objects (domain elements) and relations among them
• An atomic sentence predicate(term1,...,termn) is true
iff the objects referred to by term1,...,termn
are in the relation referred to by predicate
• Interpretation specifies referents for
constant symbols → objects
predicate symbols → relations
function symbols → functional relations
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Interpretation
12
Models for FOL: Example
13
Semantics
14
Semantics
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Universal quantification
<variables> <sentence>
All Kings are persons:
x King(x) Person(x)
x P is true in a model m
iff P is true with x being each possible object in the model
Roughly speaking, equivalent to the conjunction of instantiations of P
Richard the Lionheart is a king Richard the Lionheart is a person
King John is a king King John is a person
Richard’s left leg is a king Richard’s left leg is a person
John’s left leg is a king John’s left leg is a person
The crown is a king The crown is a person
16
A common mistake to avoid
• Typically, is the main connective with
x King(x) Person(x)
• Common mistake: using as the main connective with :
x King(x) Person(x)
means “Everyone is a king and everyone is a person”
Richard the Lionheart is a king Richard the Lionheart is a person
King John is a king King John is a person
Richard’s left leg is a king Richard’s left leg is a person
John’s left leg is a king John’s left leg is a person
The crown is a king The crown is a person
17
Existential quantification
<variables> <sentence>
x Crown(x) OnHead(x,John)
x P is true in a model m
iff P is true with x being some possible object in the model
Roughly speaking, equivalent to the disjunction of
instantiations of P
The crown is a crown the crown is on John’s head
Richard the Lionheart is a crown Richard the Lionheart is on John’s head
King John is a crown King John is on John’s head
...
18
Another common mistake to avoid
• Typically, is the main connective with
• x Crown(x) OnHead(x,John)
• Common mistake: using as the main connective with :
x Crown(x) OnHead(x,John)
is true even if there is anything which is not a crown
The crown is a crown the crown is on John’s head
Richard the Lionheart is a crown Richard the Lionheart is on John’s head
King John is a crown King John is on John’s head
19
Properties of quantifiers
x y is the same as y x, and can be written as x,y
x y is the same as y x , and can be written as x,y
x y is not the same as y x
y x Loves(x,y)– “Everyone in the world is loved by at least one person”
x y Loves(x,y)– “There is a person who loves everyone in the world”
x y P(x,y) : every object in the universe has a particular property, given by P
x y P(x,y) : there is some object in the world that has a particular property
Rule: the variable belongs to the innermost quantifier that mentions it
x [Cat(x) V (x Brother(Richard,x))]
x [Cat(x) V (z Brother(Richard,z))]
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Properties of quantifiers
• Quantifier duality: each can be expressed using the other
x Likes(x,Broccoli) = x Likes(x,Broccoli)
x Likes(x,IceCream) = x Likes(x,IceCream)
• De Morgan’s rules for quantifiers:
x P = x P
x P = x P
x P = x P
x P = x P
21
Equality
• term1 = term2 is true under a given interpretation if
and only if term1 and term2 refer to the same object
• E.g., definition of Sibling in terms of Parent:
x,y Sibling(x,y) [(x = y) m,f (m = f)
Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]
22
Writing FOL
There is somebody who is loved by everybody
23
Writing FOL
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Using FOL
The kinship domain:
• Brothers are siblings
x,y Brother(x,y) Sibling(x,y)
• One's mother is one's female parent
m,c Mother(c) = m (Female(m) Parent(m,c))
• “Sibling” is symmetric
x,y Sibling(x,y) Sibling(y,x)
• One's husband is one's male spouse
w,h Husband(h,w) (Male(m) Spouse(h,w))
• Sibling is another child of one’s parents
x,y Sibling(x,y) [(x = y) m,f (m = f) Parent(m,x)
Parent(f,x) Parent(m,y) Parent(f,y)]
25
Inference in
First Order Logic
Artificial Intelligence
Slides are mostly adapted from AIMA and MIT Open Courseware,
Milos Hauskrecht (U. Pittsburgh)
and Max Welling (UC Irvine)
26
Logical Inference
27
Inference in Propositional Logic
28
Inference in FOL : Truth Table Approach
30
Inference Rules
31
Sentences with variables
32
Sentences with variables
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Variable Substitutions
SUBST(θ, α)θ =
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Universal elimination
• Every instantiation of a universally quantified sentence is entailed by it:
v αSubst({v/g}, α)
for any variable v and ground term g
• E.g., x King(x) Greedy(x) Evil(x) yields:
King(John) Greedy(John) Evil(John), {x/John}
King(Richard) Greedy(Richard) Evil(Richard), {x/Richard}
King(Father(John)) Greedy(Father(John)) Evil(Father(John)),
{x/Father(John)}
.
.
{x/Ben}
35
Existential elimination
• For any sentence α, variable v, and constant symbol k that does notappear elsewhere in the knowledge base:
v α
Subst({v/k}, α)
• E.g., x Crown(x) OnHead(x,John) yields:
Crown(C1) OnHead(C1,John)
provided C1 is a new constant symbol, called a Skolem constant
36
Inference rules for quantifiers
αv Subst({g/v}, α)
37
Example Proof
• The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.
• Prove that Col. West is a criminal
38
Example knowledge base contd.
... it is a crime for an American to sell weapons to hostile nations:
x,y,z American(x) Weapon(y) Sells(x,y,z) Nation(z) Hostile(z) Criminal(x)
Nono … has some missiles, i.e.,
x Owns(Nono,x) Missile(x):
… all of its missiles were sold to it by Colonel West
x Missile(x) Owns(Nono,x) Sells(West,x,Nono)
Missiles are weapons:
x Missile(x) Weapon(x)
An enemy of America counts as "hostile“:
x Enemy(x,America) Hostile(x)
West, who is American …
American(West)
The country Nono
Nation(Nono)
Nono, an enemy of America …
Enemy(Nono,America), Nation(America)
39
Example knowledge base contd.
1. x,y,z American(x) Weapon(y) Sells(x,y,z) Nation(z) Hostile(z) Criminal(x)
2. x Owns(Nono,x) Missile(x):
3. x Missile(x) Owns(Nono,x) Sells(West,x,Nono)
4. x Missile(x) Weapon(x)
5. x Enemy(x,America) Hostile(x)
6. American(West)
7. Nation(Nono)
8. Enemy(Nono,America)
9. Nation(America)
10. Owns(Nono,M1) and Missile(M1) Existential elimination 2
11. Owns(Nono,M1) And elimination 10
12. Missile(M1) And elimination 10
13. Missile(M1) Weapon(M1) Universal elimination 4
14. Weapon(M1) Modus Ponens, 12, 13
15. Missile(M1) Owns(Nono,M1) Sells(West,M1,Nono) Universal Elimination 3
16. Sells(West,M1,Nono) Modus Ponens 10,15
17. American(West) Weapon(M1) Sells(West,M1,Nono) Nation(Nono) Hostile(Nono) Criminal(Nono) Universal elimination, three times 1
18. Enemy(Nono,America) Hostile(Nono) Universal Elimination 5
19. Hostile(Nono) Modus Ponens 8, 18
20. American(West) Weapon(M1) Sells(West,M1,Nono) Nation(Nono) Hostile(Nono) And Introduction 6,7,14,16,19
21. Criminal(West) Modus Ponens 17, 20
41
Reduction to propositional inference
Suppose the KB contains just the following:
x King(x) Greedy(x) Evil(x)
King(John)
Greedy(John)
Brother(Richard,John)
• Instantiating the universal sentence in all possible ways (there are only two ground terms: John and Richard) , we have:
King(John) Greedy(John) Evil(John)
King(Richard) Greedy(Richard) Evil(Richard)
King(John)
Greedy(John)
Brother(Richard,John)
• The new KB is propositionalized: proposition symbols are
King(John), Greedy(John), Evil(John), King(Richard), etc.
42
Reduction contd.
• Every FOL KB can be propositionalized so as to preserve
entailment– A ground sentence is entailed by new KB iff entailed by original KB
• Idea for doing inference in FOL:– propositionalize KB and query
– apply inference
– return result
• Problem: with function symbols, there are infinitely many
ground terms,– e.g., Father(Father(Father(John))), etc
43
Reduction contd.
Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB
Idea: For n = 0 to ∞ do
create a propositional KB by instantiating with depth-n terms
see if α is entailed by this KB
Problem: works if α is entailed, loops if α is not entailed
Theorem: Turing (1936), Church (1936)
Entailment for FOL is semidecidable (algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every nonentailed sentence.)
44
Problems with propositionalization
• Propositionalization seems to generate lots of irrelevant sentences.
• E.g., from:
x King(x) Greedy(x) Evil(x)
King(John)
y Greedy(y)
Brother(Richard,John)
• it seems obvious that Evil(John) is entailed, but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant
• With p k-ary predicates and n constants, there are p·nk instantiations.
• Lets see if we can do inference directly with FOL sentences
46
Generalized Modus Ponens (GMP)
p1', p2', … , pn', ( p1 p2 … pn q)
Subst(θ,q)
Example:
p1' is King(John) p1 is King(x)
p2' is Greedy(y) p2 is Greedy(x)
θ is {x/John, y/John} q is Evil(x)
Subst(θ,q) is Evil(John)
Example:
p1' is Missile(M1) p1 is Missile(x)
p2' is Owns(y, M1) p2 is Owns(Nono,x)
θ is {x/M1, y/Nono} q is Sells(West, Nono, x)
Subst(θ,q) is Sells(West, Nono, M1)
• Implicit assumption that all variables universally quantified
where we can unify pi‘ and pi for all i
i.e. pi'θ = pi θ for all i
GMP used with KB of definite clauses (exactly one positive literal)
47
Soundness and completeness of GMP
• Need to show that p1', …, pn', (p1 … pn q) ╞ qθ
provided that pi'θ = piθ for all I
• Lemma: For any sentence p, we have p ╞ pθ by UI
1. (p1 … pn q) ╞ (p1 … pn q)θ = (p1θ … pnθ qθ)
2. p1', \; …, \;pn' ╞ p1' … pn' ╞ p1'θ … pn'θ
3. From 1 and 2, qθ follows by ordinary Modus Ponens
GMP is sound
Only derives sentences that are logically entailed
GMP is complete for a KB consisting of definite clauses– Complete: derives all sentences that entailed
– OR…answers every query whose answers are entailed by such a KB
–
– Definite clause: disjunction of literals of which exactly 1 is positive,
e.g., King(x) AND Greedy(x) -> Evil(x)
NOT(King(x)) OR NOT(Greedy(x)) OR Evil(x)
48
Generalized Modus Ponens (GMP)
p1', p2', … , pn', ( p1 p2 … pn q)
Subst(θ,q)
Convert each sentence into cannonical form prior to inference:
Either an atomic sentence or an implication with a conjunction of
atomic sentences on the left hand side and a single atom on the right
(Horn clauses)
49
Unification
50
Unification
52
Unification
• We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y)
θ = {x/John,y/John} works
• Unify(α,β) = θ if αθ = βθ
p q θ
Knows(John,x) Knows(John,Jane) {x/Jane}}
Knows(John,x) Knows(y, Elizabeth) {x/ Elizabeth,y/John}}
Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}}
Knows(John,x) Knows(x, Elizabeth) {fail}
• Standardizing apart eliminates overlap of variables,e.g., Knows(z17, Elizabeth)
53
Unification
• To unify Knows(John,x) and Knows(y,z),
θ = {y/John, x/z } or θ = {y/John, x/John, z/John}
• The first unifier is more general than the second.
• Most general unifier is the substitution that makes the least commitment about the bindings of the variables
• There is a single most general unifier (MGU) that is unique up to renaming of variables.
MGU = { y/John, x/z }
54
The unification algorithm
55
The unification algorithm
56
Example knowledge base revisited1. x,y,z American(x) Weapon(y) Sells(x,y,z) Nation(z) Hostile(z) Criminal(x)
2. x Owns(Nono,x) Missile(x):
3. x Missile(x) Owns(Nono,x) Sells(West,x,Nono)
4. x Missile(x) Weapon(x)
5. x Enemy(x,America) Hostile(x)
6. American(West)
7. Nation(Nono)
8. Enemy(Nono,America)
9. Nation(America)
Convert the sentences into Horn form
1. American(x) Weapon(y) Sells(x,y,z) Nation(z) Hostile(z) Criminal(x)
2. Owns(Nono,M1)
3. Missile(M1)
4. Missile(x) Owns(Nono,x) Sells(West,x,Nono)
5. Missile(x) Weapon(x)
6. Enemy(x,America) Hostile(x)
7. American(West)
8. Nation(Nono)
9. Enemy(Nono,America)
10. Nation(America)
11. Proof
12. Weapon(M1)
13. Hostile(Nono)
14. Sells(West,M1,Nono)
15. Criminal(West)
57
Inference appoaches in FOL
• Forward-chaining
– Uses GMP to add new atomic sentences
– Useful for systems that make inferences as information streams in
– Requires KB to be in form of first-order definite clauses
• Backward-chaining
– Works backwards from a query to try to construct a proof
– Can suffer from repeated states and incompleteness
– Useful for query-driven inference
• Note that these methods are generalizations of their propositional equivalents
58
Forward chaining algorithm
59
Forward chaining proof
60
Forward chaining proof
61
Forward chaining proof
62
Properties of forward chaining
• Sound and complete for first-order definite clauses
• Datalog = first-order definite clauses + no functions
• FC terminates for Datalog in finite number of iterations
• May not terminate in general if α is not entailed
• This is unavoidable: entailment with definite clauses is semidecidable
63
Efficiency of forward chaining
Incremental forward chaining: no need to match a rule on iteration k if a premise wasn't added on iteration k-1
match each rule whose premise contains a newly added positive literal
Matching itself can be expensive:
Database indexing allows O(1) retrieval of known facts
– e.g., query Missile(x) retrieves Missile(M1)
Forward chaining is widely used in deductive databases
64
Backward chaining algorithm
SUBST(COMPOSE(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, p))
65
Backward chaining example
66
Backward chaining example
67
Backward chaining example
68
Backward chaining example
69
Backward chaining example
70
Backward chaining example
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Backward chaining example
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Backward chaining example
73
Properties of backward chaining
• Depth-first recursive proof search: space is linear in size of proof
• Incomplete due to infinite loops
– fix by checking current goal against every goal on stack
• Inefficient due to repeated subgoals (both success and failure)
– fix using caching of previous results (extra space)
• Widely used for logic programming
74
Logic programming: Prolog
• Algorithm = Logic + Control
• Basis: backward chaining with Horn clauses + bells & whistles
• Program = set of clauses = head :- literal1, … literaln.
criminal(X) :- american(X), weapon(Y), sells(X,Y,Z), hostile(Z).
• Depth-first, left-to-right backward chaining
• Built-in predicates for arithmetic etc., e.g., X is Y*Z+3
• Built-in predicates that have side effects (e.g., input and output
• predicates, assert/retract predicates)
• Closed-world assumption ("negation as failure")
– e.g., given alive(X) :- not dead(X).
– alive(joe) succeeds if dead(joe) fails
75
Resolution in
First Order Logic
Artificial Intelligence
Slides are mostly adapted from AIMA and MIT Open Courseware
and Milos Hauskrecht (U. Pittsburgh)
76
Resolution Inference Rule
77
First Order Resolution
78
Clausal Form
79
Converting to Clausal Form
Also move all quantifiers left
80
Converting to Clausal Form - Skolemization
81
Converting to Clausal Form - Skolemization
82
Converting to Clausal Form
83
Inference with resolution rule
84
85
Dealing with Equality
86
Example
87
More examples
88
First Order Resolution
89
Substitutions
90
Unification
91
Most General Unifier
92
Unification Algorithm
93
Unify-var subroutine
94
Examples
95
Resolution with Variables
96
Curiosity Killed the Cat
97
Proving Validity
98
Example
99
Example
100
Green’s Trick
101
Equality
102
Proof Example
103
The Clauses
104
The Query
105
The Proof
106
Hat of D
107
Who is Jane’s Lower